IAO 2006, Теория, Група α — Задача 2. False star.
Theoretical round. Problems to solve · 10 ноември 2006 г.
Бележка
Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may have no real sense. In this case you have to write in your answer (in English or Russian): «situation is impossible – ситуация невозможна». Of course, this answer has to be explained numerically or logically.
Source annotations
Printed source inconsistencies are retained: the English telescope key repeats Δx_m on the right of its displacement identity; the galaxy key uses both1.89×10⁻³ and1.86×10⁻³rad; the alpha eccentricity ratio has a sign inconsistency before the later positive0.07 result. The English polar key prints71°06′N/72°48′S; the Russian copy has handwritten70°06′N/71°48′S. No scientific correction has been silently substituted.
Условие
The White Bear (whom was already met in the VII, VIII, IX and X International Astronomy Olympiads) decided to connect a telescope to a film photo camera to make photos of constellations. The exposure times used were of order of 5 minutes. His comrade Penguin appears to be a great joker and bought a photoflash lamp. He decided to play a trick on the Bear and to “put” an extra star in his sky image, while the Bear photographed near-horizon constellations. The Penguin sitting in the photographic area of the Bear’s telescope (of course, relatively far from the telescope – 2.5 km) flashed once (sent one light pulse) to the camera during the exposure. Estimate the stellar magnitude of the false star in the Bear’s image.
Take into account the parameters of the flash pulse. The photographic conditions (sensitivity of film, diaphragm) while photographing objects with the flash at the distance of 1.5 metres (from both flash and camera) should be the same as while photographing the same objects on a sunny day with an exposure time of about 1/1000 second.
The solution has to include a picture with images of the telescope, Bear and Penguin.
Решение
Let us compare light sources: the flash-lamp and the Sun. The fact, that the object’s illumination by the flash-lamp from a distance of 1.5 meters is the same as one by the Sun during 1/1000 seconds, show that the energies received by unit of the area of photographing are equal in these cases. That is, it is possible to make a conclusion about equivalence of one flash-pulse (it is not important, how long it lasts) from distance of 1.5 m and object (stellar magnitude of the Sun), shining 1/1000 second.
But the Bear exposured the film during not 1/1000 second but 5 minutes = 300 seconds. During this time the same light energy on unit of the area will come from an object 300 000 times fainter than the Sun, that is, from object with 13.7 stellar magnitude larger, namely or of order . For comparison – a little more bright than the full Moon.
The made conclusions concern to illumination by these objects of a photographed surface. But it is possible to make the same conclusions about the shining objects themselves being under photographing. Thus, for 300 seconds exposure the flash at the distance 1.5 m would look as object .
While increasing of the distance () to the flash the energy coming to a unit of area decreases as . So increasing of the distance in 10 times leads to decreasing of light stream in 100 times ( increasing of stellar magnitude), increasing of the distance in times – in 2 800 000 times. Accordingly, “the effective stellar magnitude of the flash” will increase on 16.1 and will make . That is the Bear will find out “new object” of order at the sky image.
Оригинал в Архива: theo1.jpg · официални решения: theosol1.jpg · допълнителни данни: theo2.jpg · допълнителни данни: theodata1.pdf · допълнителни данни: theodata2.jpg · допълнителни данни: constants.jpg · допълнителни данни: theosol2.jpg · допълнителни данни: theosol3.jpg · допълнителни данни: theosol4.jpg